Notes on Logic
I'm studying for a big exam, and I've been wondering what the precise definitions are of certain terms used in logical arguments. Thanks to Merriam-Webster OnLine, I now know. And you can too! (So don't ever say you never learn anything useful at Songdog.net).
A lemma is "an auxiliary proposition used in the demonstration of another proposition"
A corollary is "a proposition inferred immediately from a proved proposition with little or no additional proof"
An axiom is "1 : a maxim widely accepted on its intrinsic merit; 2 : a statement accepted as true as the basis for argument or inference; 3 : an established rule or principle or a self-evident truth" (see also postulate)
A theorem is "1 : a formula, proposition, or statement in mathematics or logic deduced or to be deduced from other formulas or propositions; 2 : an idea accepted or proposed as a demonstrable truth often as a part of a general theory" (see also proposition)
Incidentally, a hypothesis is "1 a : an assumption or concession made for the sake of argument b : an interpretation of a practical situation or condition taken as the ground for action; 2 : a tentative assumption made in order to draw out and test its logical or empirical consequences; 3 : the antecedent clause of a conditional statement"
The difference between a theorem and a hypothesis seems subtle: a theorem is something you will attempt to prove or at least deduce, while a hypothesis is something which you may attempt to test. And therein lies the difference between the methods of mathematics and of science.
By the way, did you know that a theorem is also "3 : [a] stencil; 4 : a painting produced especially on velvet by the use of stencils for each color"?